Lie-algebra-assisted quantum simulation and quantum optimal control via high-order Magnus expansions

The evolution of a quantum system under time-dependent driving exhibits phenomena that are absent in its stationary counterpart. However, the high dimensionality and noncommutative nature of quantum dynamics make this a challenging problem. The Magnus expansion provides an analytic framework to approximate the effective dynamics on short timescales, but computing high-order terms with existing methods is computationally expensive. We introduce a scalable approach for Hamiltonians consisting of a time-independent drift term and a controllable term, in which the high-order Magnus expansion is reduced to a polynomial expression in the duration and the parameters of the time-dependent control function. For a fixed Hamiltonian model, the demanding algebraic calculation is performed once and stored in dynamical coefficients, after which the effective Hamiltonian can be evaluated numerically several orders of magnitude faster than previous techniques. We first demonstrate the method as a proof of principle for quantum simulation, verifying the expected high-order error scaling and benchmarking the numerical efficiency of the polynomial representation. We then apply the same analytically differentiable framework to a state-of-the-art quantum optimal-control problem, designing continuous control pulses for multiqubit phase gates on a neutral-atom Rydberg platform. Our results show that this method provides a practical route toward optimal control of challenging driven quantum systems.